Lemma 4.1 from [1]} says that either there is a point \(b\in B\cap [0,1]\)  so that both \(b/2\)  and \((b+1)/2\)  have distance at least \(s\)  from \(B\) , or the \(s\) -covering number of \(B\cap [0,1]\)  is comparable to \(s^{-1}\) . The former situation is called the “gap case,” while the latter is called the “dense case.”
